Link to actual problem [5566] \[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+5 \left (1+x \right ) y^{\prime }+\left (x^{2}-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
type detected by program
{"second order series method. Ordinary point", "second order series method. Taylor series method"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{i x} \operatorname {HeunC}\left (4 i, 1, -4, -2, \frac {5}{2}, \frac {x}{2}+\frac {1}{2}\right ) \left (1+x \right )}{\left (-1+x \right )^{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-i x} \left (-1+x \right )^{4} y}{\operatorname {HeunC}\left (4 i, 1, -4, -2, \frac {5}{2}, \frac {x}{2}+\frac {1}{2}\right ) \left (1+x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{i x} \operatorname {HeunC}\left (4 i, 1, -4, -2, \frac {5}{2}, \frac {x}{2}+\frac {1}{2}\right ) \left (1+x \right ) \left (\int \frac {\left (-1+x \right )^{3} {\mathrm e}^{-2 i x}}{\operatorname {HeunC}\left (4 i, 1, -4, -2, \frac {5}{2}, \frac {x}{2}+\frac {1}{2}\right )^{2} \left (1+x \right )^{2}}d x \right )}{\left (-1+x \right )^{4}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-i x} \left (-1+x \right )^{4} y}{\operatorname {HeunC}\left (4 i, 1, -4, -2, \frac {5}{2}, \frac {x}{2}+\frac {1}{2}\right ) \left (1+x \right ) \left (\int \frac {\left (-1+x \right )^{3} {\mathrm e}^{-2 i x}}{\operatorname {HeunC}\left (4 i, 1, -4, -2, \frac {5}{2}, \frac {x}{2}+\frac {1}{2}\right )^{2} \left (1+x \right )^{2}}d x \right )}\right ] \\ \end{align*}